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Find Fit second degree parabola equation {{1996,1997,1998,1999,2000},{40,50,62,58,60}} [ Calculator, Method and examples ]

Solution:
Your problem -> Fit second degree parabola equation {{1996,1997,1998,1999,2000},{40,50,62,58,60}}

The equation is y = a + bx + cx^2 and the normal equations are

sum y = an + b sum x + c sum x^2

sum xy = a sum x + b sum x^2 + c sum x^3

sum x^2y = a sum x^2 + b sum x^3 + c sum x^4

 X y x = X - 1998 x^2 x^3 x^4 x*y x^2*y 1996 40 -2 4 -8 16 -80 160 1997 50 -1 1 -1 1 -50 50 1998 62 0 0 0 0 0 0 1999 58 1 1 1 1 58 58 2000 60 2 4 8 16 120 240 --- --- --- --- --- --- --- --- 9990 270 0 10 0 34 48 508

Substituting these values in the normal equations
270=5a+0b+10c

48=0a+10b+0c

508=10a+0b+34c

Solving these 3 equations using inverse matrix method,
Here 5a+10c=270
10b=48
10a+34c=508

Now converting given equations into matrix form
[[5,0,10],[0,10,0],[10,0,34]] [[a],[b],[c]]=[[270],[48],[508]]

Now, A = [[5,0,10],[0,10,0],[10,0,34]], X = [[a],[b],[c]] and B = [[270],[48],[508]]

:. AX = B

:. X = A^-1 B

|A| =
 5 0 10 0 10 0 10 0 34

=
5 ×
 10 0 0 34
+0 ×
 0 0 10 34
+10 ×
 0 10 10 0

=5 xx (10 × 34 - 0 × 0) +0 xx (0 × 34 - 0 × 10) +10 xx (0 × 0 - 10 × 10)

=5 xx (340 +0) +0 xx (0 +0) +10 xx (0 -100)

=5 xx (340) +0 xx (0) +10 xx (-100)

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